Having the most general Lagrangian of the Proca Field given by $$\mathcal{L}=C_1(\partial_\nu A_\mu)(\partial^\nu A^\mu)+C_2(\partial_\nu A_\mu)(\partial^\mu A^\nu)+C_3 A_\mu A^\mu$$ the canonical momentum can be calculated by $$\Pi_\nu\equiv\frac{\partial\mathcal{L}}{\partial(\partial_0 A ^\nu)}=2C_1\partial^0 A_\nu+2C_2\partial_\nu A^0.$$ I have been trying to express the Hamiltonian density $\mathcal{H}$ only as momenta and the fields (so without any field derivatives) as I know it should be possible. It is given by $$\mathcal{H}=\Pi_\nu \partial_0A^\nu-\mathcal{L}$$ I have tried expressing the whole thing in non-covariant form and so far I have the following: $$\mathcal{H}=C_4\Pi_0\Pi_0+C_1\Big((\partial_i A_0)(\partial_i A_0)-(\partial_0 A_i)(\partial_0 A_i)\Big)-(C_1+C_2)(\partial_i A_j)(\partial_i A_j)-C_3A_0A_0+C_3A_iA_i$$ I am not 100% sure that this is correct so far but I'm pretty sure. How can I get rid of the rest of the derivative terms using the canonical momenta?


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